Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions

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Chapter 2 Polynomial and Rational Functions 2.1
Quadratic Functions

• Definition of a polynomial function
• Let n be a nonnegative integer so n0,1,2,3
• Let be real
  numbers with
• The function given by
• Is called a polynomial function of x with degree
  n
• Example
• This is a 4th degree polynomial

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Polynomial Functions are classified by degree

• For example
• In Chapter 1
• Polynomial function
• , with
• Example
• This function has
• degree 0, is a
• horizontal line and is called
• a constant function.

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Polynomial Functions are classified by degree

• In Chapter 1
• A Polynomial function
• ,
• is a line whose slope is m
• and y-intercept is (0,b)
• Example
• This function has a degree
• of 1,and is called a
• linear function.

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Section 2.1 Quadratic Functions

• Definition of a quadratic function
• Let a, b, and c be real numbers with .
• The function given by f(x)
• Is called a quadratic function
• This is a special U shaped curve called a ?

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Parabola !

• Parabolas are symmetric to a line called the axis
  of symmetry.
• The point where the axis intersects with the
  parabola is the vertex.

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The simplest type of quadratic is

• When sketching
• Use as a reference.
• (This is the simplest type of graph)
• agt1 the graph of yaf(x)
• is a vertical stretch of the
• graph yf(x)
• 0ltalt1 the graph of yaf(x)
• is a vertical shrink of the graph yf(x)
• Graph on your calculator
• , ,

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Standard Form of a quadratic Function
The graph of f(x) is a parabola whose axis is the
vertical line xh and whose vertex is the point (
, ). -shifts the graph right or
left -shifts the graph up or
down For agt0 the parabola opens up alt0 the
parabola opens down
NOTE!
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Example of a Quadratic in Standard Form

• Graph
• Where is the Vertex? ( , )
• Graph
• Where is the Vertex? ( , )

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Identifying the vertex of a quadratic function

• One way to find the vertex is to put the
  quadratic function in standard form by completing
  the square.
• Where is the vertex? ( , )

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Identifying the vertex of a quadratic function

• Another way to find the vertex is to use
• the Vertex Formula
• If agt0, f has a minimum x
• If alt0, f has a maximum x
• a b c
• NOTE
  the vertex is ( , )
• To use Vertex Formula-
• To use completing the square start
• with to get
  

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Identifying the vertex of a quadratic
function(Example)

• Find the vertex of the parabola ( , )
• The direction the parabola opens?________
• By completing the square? By the Vertex Formula

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Identifying the x-Intercepts of a quadratic
function

• The x-intercepts are found as follows

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Identifying the x-Intercepts of a quadratic
function (continued)

• Standard form is
• Shape_______________
• Opens up or down?_____
• X-intercepts are

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Identifying the x-Intercepts of a Quadratic
Function (Practice)

• Find the x-intercepts of

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Writing the equation of a Parabola in Standard
Form

• Vertex is
• The parabola passes through point
• Remember the vertex is
• Because the parabola passed through we
  have

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Writing the equation of a Parabola in Standard
Form (Practice)

• Vertex is
• The parabola passes through point
• Find the Standard Form of the equation.

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Homeworkp.95-96 1-8 all, 9-33x3

• p. 96 36-60x3